Effective Modes for a Strongly Coupled Quantum Emitter-MoS$_{2}$ Nanodisk System

We present an effective modes theory for studying the strong light-matter interaction and better understanding the non-Markovian dynamics of a quantum emitter (QE) coupled to two-dimensional materials. Specifically, we investigate the spontaneous emission of a V-type QE in close proximity to a MoS $_{2}$ nanodisk. The non-Markovian population dynamics is observed and found linked to the initial states, indicating that the coupled system enters the strong-coupling regime. The effective modes analysis shows that though a dozen of resonant modes appear in the photonic density of states, only a few modes have significant contribution to the population dynamics. Our work demonstrates the effective modes theory as a powerful tool to explore and optimize the quantum states control of QE by two-dimensional materials.